The number of electrons in the shell equals 2n² and in each subshell is 2(2l + 1), derived from the quantum mechanical principles and the Pauli exclusion principle governing electron arrangement in atoms.
Explanation:To prove that the number of electrons in the shell equals 2n² and that the number in each subshell is 2(2l + 1), we need to use the quantum mechanical principles that govern the arrangement of electrons in an atom. According to the quantum model, each electron in an atom is described by four quantum numbers: n, l, mi, and ms. The principal quantum number n represents the shell level, the angular momentum quantum number l describes the subshell (with values ranging from 0 to n-1), the magnetic quantum number mi describes the orientation of the subshell (ranging from -l to +l), and the spin quantum number ms indicates the electron's spin (which can be +1/2 or -1/2).
Each shell level n can have subshell values from 0 to n-1. For each value of l, there are 2l+1 possible values for mi, and for each mi, there can be two electrons (one with spin up and one with spin down, according to the Pauli exclusion principle). Therefore, the maximum number of electrons in any subshell is 2(2l+1). Summing over all values of l will give the total number of electrons in a shell, which is 2n². This follows from considering all possible orientations and spin states for each value of l within a shell. For the n=2 shell as an example, there are l=0 and l=1 subshells. The s subshell (l=0) can hold 2 electrons, and the p subshell (l=1) can hold 6 electrons, for a total of 2(2^2) = 8 electrons in the n=2 shell. Using similar calculations for other values of n will confirm the general formula.
The application of the Pauli exclusion principle ensures that no two electrons can have the same set of all four quantum numbers, which fundamentally limits the number of possible electrons in a subshell and a shell.
Find the X intercepts of the parabola with the vertex (1,-9) and y intercept of (0,-6)
The x-intercepts of the parabola with vertex (1,-9) and y-intercept of (0,-6) are of [tex]x = 1 \pm \sqrt{3}[/tex].
What is the equation of a parabola given it’s vertex?The equation of a quadratic function, of vertex (h,k), is given by:
y = a(x - h)² + k
In which a is the leading coefficient.
In this problem, the parabola has vertex (1,-9), hence h = 1, k = -9, and:
y = a(x - 1)^2 - 9.
The y-intercept is of (0,-6), hence when x = 0, y = -6, and this is used to find a.
-6 = a - 9
a = 3.
So the equation is:
y = 3(x - 1)^2 - 9.
y = 3x² - 6x - 6.
The x-intercepts are the values of x for which:
3x² - 6x - 6 = 0.
Then:
x² - 2x - 2 = 0.
Which has coefficients a = 1, b = -2, c = -2, hence:
[tex]\Delta = b^2 - 4ac = (-2)^2 - 4(1)(-2) = 12[/tex]
[tex]x_1 = \frac{2 + \sqrt{12}}{2} = 1 + \sqrt{3}[/tex]
[tex]x_2 = \frac{2 - \sqrt{12}}{2} = 1 - \sqrt{3}[/tex]
The x-intercepts of the parabola are [tex]x = 1 \pm \sqrt{3}[/tex].
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Suppose a study estimated that 85% of the residents of a town (with an error range of ±12 percentage points at 95% confidence) favor building a new community center. Which of the following percentages of the town's residents may favor building a new community center?
A. 69%
B. 79%
C. 59%
D. 99%
The confidence interval for a given sample value can be calculated using the following formula:
Confidence interval = Average value ± Margin of error
Which in this case the values are:
Average value = 85%
Margin of error = 12%
Therefore substituting the given values into the equation will give us:
Confidence interval = 85 ± 12
Confidence interval = 73, 97
Therefore the percentage of the residents of the town who are favour of building a new community center ranges from 73% to 97%.
Based from the given choices, only letter B 79% is within this range:
Answer:
B. 79%
Answer:
B. 79%
Step-by-step explanation:
yes
Find the area of a regular hexagon whose side length is 16 in. and the apothem is 8 square root of 3 in
just need to check these answers
2 Questions~
Evaluate the expression.
38+16⋅12÷2−(30⋅2)
Simplify this expression.
5 + 10 ÷ 5
3
7
10
15
Your answer would be a.
Answer: 3
hope this helps! :)
~Izzie
Suppose you buy a CD for $500 that earns 2.5% APR and is compounded quarterly. The CD matures in 3 years. How much will the CD be worth at maturity?
8% of x is equal to 48
divide 48 by 8%
48/0.08 = 600
check
600*0.08 = 48
x=600
$35,485.00 to $50,606.00 per year is equivalent to how much an hour
assuming it is based on a 40 hour work week, working 52 weeks per year:
35485/52 = 682.40 per week
682.40/40 = 17.06 per hour
50606/52 = 973.19 per week
973.19/40 = 24.33 per hour
so between 17.06 & 24.33 per hour
What is the 5th term of an arithmetic sequence if t3 = 10 and t7 = 26?
18
20
22
24
which one is it? need help please
Mary, who is sixteen years old, is four times as old as her brother. how old will mary be when she is twice as old as her brother? explained
Use basic identities to simplify the expression. sin2θ + tan2θ + cos2θ
Can anyone please help ASAP, will give thanks and all that fun stuff
Answer:
The answer is B!!!!
Answers for 1.2.1 how can I describe a graph
The domain of the function is given. Find the range.
f(x) = 2x - 1
Domain: {-2, 0, 2, 4}
25 decreased by 1/5 of a number is 18
The equation for the student's question is 25 - (1/5)x = 18. Solving for x involves simple algebraic manipulation, resulting in x being equal to 35.
Explanation:The student's question '25 decreased by 1/5 of a number is 18' is a basic algebra problem. We could represent the unknown number as x. So the equation would be 25 - (1/5)x = 18.
To solve the equation 25 decreased by 1/5 of a number is equal to 18, we can set up the equation as 25 - (1/5)x = 18, where x is the unknown number.
To isolate x, we first subtract 25 from both sides of the equation:
- (1/5)x = -7.
Next, we can multiply both sides of the equation by -5 to eliminate the fraction:
x = (-7) * (-5) = 35.
To solve for x, first, add (1/5)x to both sides to get 25 = 18 + (1/5)x.
Then, subtract 18 from both sides to obtain 7 = (1/5)x. Finally, multiply both sides by 5 to find the value of x. Thus, x equals 35.
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A phone company offers two monthly plans. Plan A costs $30 plus an additional $0.15 for each minute of calls. Plan B costs $16 plus an additional $0.20 for each minute of calls.
For what amount of calling do the two plans cost the same?
What is the cost when the two plans cost the same?
Plan A = 30 +0.15x
Plan B = 16 +0.20x
30+0.15x = 16+0.20x
subtract 16 from each side
14 +0.15x = 0.20x
subtract 0.15x from each side
14=0.05x
x = 14/0.05 = 280 minutes
280*0.15 = 42 +30 = $72
280 * 0.20 = 56 +16 = 72
280 minutes and cost $72 each
Which of the following is true when probability answer is written in the form of a fraction
In probability, a fraction does not need to be simplified but converting to a percentage or decimal can aid in interpretation, with percentages being fractions with a denominator of 100. (First option)
When a probability answer is written in the form of a fraction, it is important to know how to handle the result. Simplifying the fraction is not generally required, especially in Probability Topics, where the focus is on understanding the probability itself rather than the arithmetic of fractions.
However, converting the probability to a percentage is a common practice that can help with interpretation. This is done by writing the value of the percent as a fraction with a denominator of 100 and then simplifying it if possible.
Converting a fraction to a decimal is another common step, which involves dividing the numerator by the denominator. It's also important to note that for rounding answers to probability problems, the convention is to round to four decimal places.
JL is a common tangent to circles M and K at point J. If angle MLK measures 61ᵒ, what is the length of radius MJ? Round to the nearest hundredth. (Hint: Show that triangles LMJ and LKJ are right triangles, and then use right triangle trigonometry to solving for missing sides of the right triangles.)
To find the length of radius MJ, we can use right triangle trigonometry. Firstly, we can show that triangles LMJ and LKJ are right triangles. Then, we can use the given angle MLK of 61ᵒ to find the length of radius MJ, using the sine function. The equation to find MJ is MJ = rM * sin(29ᵒ).
Explanation:To find the length of radius MJ, we can use right triangle trigonometry. Firstly, we can show that triangles LMJ and LKJ are right triangles. Since JL is a common tangent, it is perpendicular to the radii of the circles at points J. Therefore, angle LMJ and angle LKJ are right angles. Now, we can use the given angle MLK of 61ᵒ to find the length of radius MJ.
Let's call the radius of circle M rM and the radius of circle K rK. In triangle LMJ, we have the following relationships:
angle LMJ = 90ᵒ (since it is a right triangle)angle MLJ = angle MLK - angle JLK = 61ᵒ - 90ᵒ = -29ᵒ (since angle JLK is a right angle)angle MJL = angle JML = 90ᵒ - angle MLJ = 90ᵒ - (61ᵒ - 90ᵒ) = 119ᵒUsing the sine function, we can find the length of side MJ:
sin(angle MLJ) = length of side MJ / length of side LJ
sin(-29ᵒ) = MJ / rM
Since sin(angle MLJ) = -sin(angle MJL), we can rewrite the equation as:
sin(29ᵒ) = MJ / rM
Now, we can rearrange the equation to solve for MJ:
MJ = rM * sin(29ᵒ)
Since we are not given the values of rM or rK, we cannot find the specific value of MJ. However, we can use this equation to find the length of radius MJ if we are given the values of the radii of the circles and the given angle MLK.
Remember to round the answer to the nearest hundredth as specified in the question.
The cost to produce a product is modeled by the function f(x) = 5x2 − 70x + 258 where x is the number of products produced. Complete the square to determine the minimum cost of producing this product.
The minimum cost of producing this product is:
13
Step-by-step explanation:The function which is used to represent the cost to produce x elements is given by:
[tex]f(x)=5x^2-70x+258[/tex]
Now, on simplifying this term we have:
[tex]f(x)=5(x^2-14x)+258\\\\i.e.\\\\f(x)=5(x^2+49-49-14x)+258\\\\i.e.\\\\f(x)=5((x-7)^2-49)+258\\\\i.e.\\\\f(x)=5(x-7)^2-5\times 49+258\\\\i.e.\\\\f(x)=5(x-7)^2-245+258\\\\i.e.\\\\f(x)=5(x-7)^2+13[/tex]
We know that:
[tex](x-7)^2\geq 0\\\\i.e.\\\\5(x-7)^2\geq 0\\\\i.e.\\\\5(x-7)^2+13\geq 13[/tex]
This means that:
[tex]f(x)\geq 13[/tex]
This means that the minimum cost of producing this product is: 13
Explain how the GCF helps with the distributive property. Why is it so important to use the GCF when factoring a sum of two numbers?
GCF is the greatest common factor, that divides two number and the distributive property is that when a number multiplied with each number in the bracket and then perform addition or subtraction etc.
GCF help with distributive property and it is important to use when factoring a sum of two numbers. For example we have to add fractions 2/3 and 4/9, now the GCF is 9
2/3 + 4/9
= 2(3) + 4 (1) / 9
Now 2 is the common factor, so it allow us to use the distributive property.
= 2 (3 + 2) / 9
= 2(5) /9
=10 / 9 is the answer.
Find the coordinates of point Q that lies along the directed line segment from R(-2, 4) to S(18, -6) and partitions the segment in the ratio of 3:7.
Answer:
(2 2/7, 5 1/3)
Step-by-step explanation:
The coordinates of point Q, lies along R(-2,4) and S(18,-6)
thus, QR and RS, that is in ratio of QR : RS = 3 : 7
Let point Q = (x,y)
Hence, QR = -2 - x; RS = -6 - 4
Thus, QR/RS = 3/7, which is: (-2 - x)/(-6 - 4) = 3/7
7(-2 - x) = -30
-14 - 7x = -30
7x = 16
∴ x = 16/7 = 2 2/7
If x : y = 3 : 7 ( where x = 2 2/7)
Hence, (2 2/7)/y = 3/7
3y = 16
∴ y = 16/3 = 5 1/3
The coordinates of point Q = (2 2/7, 5 1/3)
In a batch of 280 water purifiers, 12 were found to be defective. What is the probability that a water purifier chosen at random will be defective? Write the probability as a percent. Round to the nearest tenth of a percent if necessary
Answer:
[tex]\text{Probability}=4.3\%[/tex]
Step-by-step explanation:
Given : In a batch of 280 water purifiers, 12 were found to be defective.
To find : What is the probability that a water purifier chosen at random will be defective? Write the probability as a percent.
Solution :
Total number of batch of purifiers = 280
Number of defective purifiers = 12
The probability that a water purifier chosen at random will be defective is given by,
[tex]\text{Probability}=\frac{\text{Favorable outcome}}{\text{Total outcome}}[/tex]
[tex]\text{Probability}=\frac{12}{280}[/tex]
[tex]\text{Probability}=\frac{3}{70}[/tex]
Converting into percentage,
[tex]\text{Probability}=\frac{3}{70}\times 100[/tex]
[tex]\text{Probability}=4.28\%[/tex]
Round to nearest tenths,
[tex]\text{Probability}=4.3\%[/tex]
Stanfing in your tree house 50ft off the ground you look down at a 60 degree angle how many feet from the base of the tree is the frisbee
multiply 50 x tan(60) = 86.6 feet.
round your answer as needed
The point (–3, –5) is on the graph of a function. Which equation must be true regarding the function?
f(–3) = –5
f(–3, –5) = –8
f(–5) = –3
f(–5, –3) = –2
Answer:
Option 1st is correct
[tex]f(-3) = -5[/tex]
Step-by-step explanation:
If any point [tex](x, y)[/tex] is on the graph then we can write the function as:
[tex]y= f(x)[/tex]
where
x is the independent variable and
y is the dependent variable.
As per the statement:
The point (–3, –5) is on the graph of a function.
⇒x = -3 and y = -5
By above definition we have;
[tex]f(-3) = -5[/tex]
Therefore, the equation must be true regarding the function is, [tex]f(-3) = -5[/tex]
A doorway is 8 feet high and 4 feet wide. A square piece of plywood needs to be moved through the doorway. The plywood is 10 feet long and 10 feet wide. The door is a rectangle with a height of 8 feet, and a width of 4 feet. A dotted line shows the diagonal.
Will the piece of plywood fit through the door if it is tilted diagonally?
A. No, because the length of the diagonal is close to 9 feet.
B. No, because the height of the door is less than 10 feet.
C. Yes, because the length of the diagonal is close to 11 feet.
D. Yes, because the sum of the height and width of the door is greater than 10 feet.
After calculating the diagonal of the doorway to be approximately 8.944 feet using the Pythagorean theorem, it's clear that the 10-foot square piece of plywood will not fit diagonally through the door. Thus, the correct answer is option (A).
The question is whether a 10-foot square piece of plywood can fit through an 8-foot by 4-foot doorway when tilted diagonally. To determine if the plywood can fit, we need to calculate the diagonal of the doorway using the Pythagorean theorem which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as c² = a² + b² where c is the hypotenuse, and a and b are the other two sides.
Let's apply the theorem to our doorway:
Height (a) = 8 feet
Width (b) = 4 feet
Diagonal (c) = ?
We calculate the diagonal:
c² = a² + b²
c² = 8² + 4²
c² = 64 + 16
c² = 80
c = sqrt(80)
c = 8.944 feet (approx)
The diagonal of the doorway is approximately 8.944 feet, which is less than the 10 feet length of the plywood. Therefore, the correct answer is:
No, because the length of the diagonal is close to 9 feet.
The option (A) is correct.
What are the exact solutions of x2 − 3x − 1 = 0?
Select one:
a. x = the quantity of 3 plus or minus the square root of 5 all over 2 Incorrect
b. x = the quantity of negative 3 plus or minus the square root of 5 all over 2
c. x = the quantity of 3 plus or minus the square root of 13 all over 2
d. x = the quantity of negative 3 plus or minus the square root of 13 all over 2
Why do we state restrictions for rational expression and when do we state the restrictions?
We state restrictions for rational expressions to ensure that the denominator does not equal zero, as division by zero is undefined in mathematics. The restrictions are the values of the variable that make the denominator equal to zero. We state the restrictions whenever we are simplifying, performing operations with, or solving rational expressions.
A rational expression is an expression that can be written in the form of a fraction, where the numerator and the denominator are polynomials. The denominator of a rational expression cannot be zero because division by zero is not defined in mathematics. Therefore, when working with rational expressions, it is crucial to identify the values of the variable that would make the denominator equal to zero. These values are the restrictions, or domain restrictions, for the rational expression.
For example, consider the rational expression [tex]\(\frac{1}{x-3}\)[/tex]. The denominator is[tex]\(x-3\)[/tex]. To find the restriction, we set the denominator equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[x - 3 = 0\][/tex]
[tex]\[x = 3\][/tex]
Therefore, the restriction for this rational expression is [tex]\(x \neq 3\)[/tex], meaning that [tex]\(x\)[/tex] can be any real number except 3.
We must state these restrictions whenever we perform operations such as simplifying, adding, subtracting, multiplying, or dividing rational expressions, as well as when we are solving rational equations. This ensures that the operations are valid and that the solutions to the equations do not include any undefined expressions.
In summary, stating restrictions for rational expressions is a critical step in avoiding mathematical errors and ensuring that the expressions and equations we work with are well-defined.
A Ferris wheel has a diameter of 42 feet. It rotates 3 times per minute. Approximately how far will a passenger travel during a 5-minute ride?
132 feet
659 feet
1,978 feet
3,956 feet
Approximately 1978 ft a passenger travel during a 5-minute ride and this can be determined by using the formula of the perimeter of a circle.
Given :
A Ferris wheel has a diameter of 42 feet. It rotates 3 times per minute.
The following steps can be used in order to determine the total distance travel by the passenger during a 5 minutes ride:
Step 1 - First determine the perimeter of the circle. The formula of the perimeter of the circle is given by:
[tex]\rm C = 2\pi r[/tex]
Step 2 - Now, substitute the value of known terms in the above formula.
[tex]\rm C=2\pi\times(21)[/tex]
[tex]\rm C= 131.94\;ft[/tex]
Step 3 - In one minute passenger travels:
[tex]\rm =131.94\times 3=395.82\; ft[/tex]
Step 4 - So, in three minutes passenger travels:
[tex]=395.82\times5[/tex]
= 1978 ft
So, approximately 1978 ft a passenger travel during a 5-minute ride.
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8x-2y over 10xy if x=4 and y=-7
The value of the expression 8x-2y over 10xy when x=4 and y=-7 is -0.1643 (rounded to four decimal places).
Explanation:To evaluate the expression 8x-2y over 10xy when x=4 and y=-7, we substitute these values into the expression:
8(4)-2(-7) over 10(4)(-7)
Simplifying further,
32+14 over -280
46 over -280
Therefore, the value of the expression 8x-2y over 10xy when x=4 and y=-7 is -0.1643 (rounded to four decimal places).
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